3-1 Lumped System Analysis 3- Nondimensional Heat Conduction Equation 3-3 Transient Heat Conduction in Semi-Infinite Solids 3-4 Periodic Heating Y.C. Shih Spring 009
3-1 Lumped System Analysis (1) In heat transfer analysis, some bodies are essentially isothermal and can be treated as a lump system. An energy balance of an isothermal solid for the time interval dt can be expressed as A s Heat Transfer into the body during dt = The increase in the energy of the body during dt h SOLID BODY m=mass V=volume ρ=density T i =initial temperature T=T(t) T () Q& = has T T t ha s (T -T)dt= mc p dt Y.C. Shih Spring 009 3-1
3-1 Lumped System Analysis () d( T T ) has = dt T T ρvcp Integrating from time zero (at which T=T i ) to t gives Tt () T has ln = t Ti T ρvcp Taking the exponential of both sides and rearranging Tt () T bt has = e ; b= (1/s) T T ρvc i b is a positive quantity whose dimension is (time) -1, and is called the time constant. p (1) 3- Y.C. Shih Spring 009
3-1 Lumped System Analysis (3) There are several observations that can be made from this figure and the relation above: 1. Equation (1) enables us to determine the temperature T(t) of a body at time t, or alternatively, the time t required for the temperature to reach a specified value T(t).. The temperature of a body approaches the ambient temperature T exponentially. 3. The temperature of the body changes rapidly at the beginning, but rather slowly later on. 4. A large value of b indicates that the body approaches the ambient temperature in a short time. 3-3 Y.C. Shih Spring 009
3-1 Lumped System Analysis (4) Criteria for Lumped System Analysis Assuming lumped system is not always appropriate, the first step in establishing a criterion for the applicability is to define a characteristic length Bi and a Biot number (Bi) as It can also be expressed as Lc = k Rcond 1 = R = conv h L c = V A s Bi = hl c Conduction resistance within the body Convection resistance at the surface of the body k T R conv h T s R cond T in 3-4 Y.C. Shih Spring 009
3-1 Lumped System Analysis (5) Lumped system analysis assumes a uniform temperature distribution throughout the body, which is true only when the thermal resistance of the body to heat conduction is zero. The smaller the Bi number, the more accurate the lumped system analysis. It is generally accepted that lumped system analysis is applicable if Bi 0.1 3-5 Y.C. Shih Spring 009
3- Nondimensional Heat Conduction Equation (1) One-dimensional transient heat conduction equation problem (0 x L): T 1 T Differential equation: = x α t Boundary conditions: Initial condition: ( 0, t) T = 0 x T( L, t) k = h T( L, t) T x ( ) T x,0 = Ti 3-6 Y.C. Shih Spring 009
3- Nondimensional Heat Conduction Equation () A dimensionless space variable X=x/L A dimensionless temperature variable θ(x, t)=[t(x,t)-t ]/[T i -T ] The dimensionless time and h/k ratio will be obtained through the analysis given below Introducing the dimensionless variable θ θ θ θ = = = = X x L T T x X T T x t T T t θ L T θ θ( 1, t) hl θ( 0, t) = ; = θ ( 1, t) ; = 0 X α x t X k X L T L T 1 T ; ; ( / ) i i i Y.C. Shih Spring 009 3-7
3- Nondimensional Heat Conduction Equation (3) Therefore, the dimensionless time is τ=αt/l, which is called the Fourier number (Fo). hl/k is the Biot number (Bi). The one-dimensional transient heat conduction problem in a plane wall can be expressed in nondimensional form as Differential equation: Boundary conditions: Initial condition: θ θ = X τ θ( 0, τ) X θ X ( 1, τ) = 0 = Biθ θ ( X,0) = 1 ( 1, τ ) 3-8 Y.C. Shih Spring 009
3- Nondimensional Heat Conduction Equation (4) Fourier number: αt kl ( 1/ L) ΔT τ = = = L c L / t ΔT ρ 3 p The rate at which heat is conducted across L of a body of volume L 3 The rate at which heat is stored in a body of volume L 3 The Fourier number is a measure of heat conducted through a body relative to heat stored. A large value of the Fourier number indicates faster propagation of heat through a body. 3-9 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (1) A semi-infinite solid is an idealized body that has a single plane surface and extends to infinity in all directions. Assumptions: constant thermophysical properties no internal heat generation uniform thermal conditions on its exposed surface initially a uniform temperature of T i throughout. Heat transfer in this case occurs only in the direction normal to the surface (the x direction) one-dimensional problem. 3-10 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids () Differential equation: Boundary conditions: T 1 T = x α t ( 0, ) s (, ) T t = T T x t = Ti Initial condition: ( ) T x,0 = Ti The separation of variables technique does not work in this case since the medium is infinite. The partial differential equation can be converted into an ordinary differential equation by combining the two independent variables x and t into a single variable h, called the similarity variable. Y.C. Shih Spring 009 3-11
3-3 Transient Heat Conduction in Semi-Infinite Solids (3) Similarity Solution: For transient conduction in a semi-infinite medium Similarity variable: η = x 4αt Assuming T=T(h) (to be verified) and using the chain rule, all derivatives in the heat conduction equation can be transformed into the new variable T x = 1 T α t T η = T η η 3-1 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (4) T T = η η η Noting that h=0 at x=0 and h as x (and also at t=0) and substituting into Eqs. 4 37b (BC) give, after simplification T 0 = T ; T η = T ( ) ( ) s Note that the second boundary condition and the initial condition result in the same boundary condition. Both the transformed equation and the boundary conditions depend on h only and are independent of x and t. Therefore, transformation is successful, and h is indeed a similarity variable. i 3-13 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (5) Let w=dt/dh. dw dw = ηw = ηdη ln( w) = η + C dη w where C 1 =ln(c 0 ). Back substituting w=dt/dh and integrating again, w= Ce η 1 where u is a dummy integration variable. The boundary condition at h=0 gives C =T s, and the one for h gives 0 η T = C e du+ C u 1 0 u π 1 1 s 1 0 Ti = C e du+ C = C + T C = ( T T ) i π s 3-14 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (6) T T i T s T s η u = e du = erf = erfc π 0 ( η ) 1 ( η ) Where η u u erf ( η) = e du ; erfc( η) = 1 e du π 0 π 0 are called the error function and the complementary error function, respectively, of argument h. η 3-15 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (7) Knowing the temperature distribution, the heat flux at the surface can be determined from the Fourier s law to be η 1 x= 0 η= 0 4 η= 0 ( T) T T η 1 kts q& s = k = k = kce = x η x αt παt i 3-16 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (8) Other Boundary Conditions: Analytical solutions can be obtained for other boundary conditions on the surface and are given in the book Specified Surface Temperature, T s = constant. Constant and specified surface heat flux. Convection on the Surface, 3-17 Y.C. Shih Spring 009
3-3 Transient Heat Conduction in Semi-Infinite Solids (9) Application: Solidification x T = M + αt ( T T0 ) erfc T0 3-18 Y.C. Shih Spring 009
3-4 Periodic Heating 3-19 Y.C. Shih Spring 009